Many model-based approaches to the analysis of the complicated motion of an object, such as the left ventricle (LV) of the heart, and the quantification of its measured deformation, exist. Simple analytical models like spheres, ellipsoids or cylinders often are used. For example, Beyar and Sideman constructed a mechanical model where the LV is modeled as a nested-shell spheroidal to explain the effect of twisting motion; Arts (1982) used a thick-walled cylinder composed of eight concentric cylinder shells to describe the LV deformation; Kim et al. used a thick-walled ellipsoidal model for the computation of the LV wall stress, and a truncated ellipsoidal model for simulation of regional stress; Azhari et al. used a hollow conical shell (with a constant shear modulus) to characterize transmural twist motion; and Arts (1992, 1993) developed a kinematic model by combining the deformation modes from a spherical, a prolate ellipsoidal and a cylindrical model. However, the shape of the L is neither spherical nor cylindrical. Even a prolate ellipsoid, is a gross simplification of the shape of an LV. Therefore, the analyses made by these models make simplifying assumptions about the material behavior of the heart muscle and the governing equations of motion.
Recently, techniques based on the use of deformable models for reconstructing the 3D surface shape and motion of the L from CT or MRI data have been developed. The techniques use finite elements, spring-mass systems, deformation modes, bending and stretching thin-plate models, and other physics-based or geometric techniques to fashion the desirable model. The main limitation of most techniques is that they do not provide intuitive motion parameters to model the rigid and non-rigid motion of the LV. Most of the techniques represent the L motion as a set of local displacement vectors which either requires non-trivial post-processing to be useful to a physician or it is only good for qualitative visualization.
On the other hand, some models tend to be formulated in terms of very few parameters that can offer only a gross approximation to the motion of the LV. Attempts to characterize the L motion based on deformation modes, also do not provide a suitable localization of the L deformations and motion in a clinically useful manner without complex post-processing, due to the global definition of each modal deformation. Moreover, most techniques ignore the twisting or wringing motion of the LV known to occur during systole. A class of surface-deformable primitives whose global parameters are functions were developed to overcome the existing model limitations. The utility of this class, however, is usually limited to representing LV surface shape and motion because the L motion cannot be captured entirely with surface models.
To capture the LV shape and motion throughout its volume, a volumetric deformable model is desirable. Recently, techniques for analyzing the L volumetric shape and motion from tagged MR image sequences have also been developed based on the use of 3D finite elements. Such model-based approaches to the recovery of the volumetric shape and motion of the LV can be useful in overcoming the limitation of current medical imaging modalities. Such modalities typically cannot provide explicit, time-varying 3D motion of material data points from to the LV. The translation of the deforming heart through the fixed image planes puts a different cross section of myocardium into the image plane at each time point. Therefore the imaging techniques can provide only the two-dimensional motion of the object's material points on the imaging plane at best.
Finite element modeling is a typical choice for volumetric motion analysis, because it provides strain analysis throughout the LV wall. However, the finite element representation does not directly lend itself to an understanding of the underlying kinematics in a clinically useful way. The parameters of a finite element model are nodal displacements which can result in a relatively large number of parameters. The physical interpretation of these parameters can be difficult. For example, the 3D strain tensor has three normal components and three shear components, each of which may vary with position in the LV wall. In order to understand the complex relationship between these components and other motion parameters, it is desirable to characterize the motion in terms of meaningful physical parameters-that offer sufficient accuracy.
Characterization of heart wall motion on a regional level is desired to understand cardiac mechanics and the processes underlying a disease. In order to accurately measure heart wall motion, a number of material points in the heart must be located and tracked. Methods for providing intra-myocardial markers in the past have included the implantation of radiopaque markers, lead beads or ultrasonic crystals, use of naturally occurring landmarks, and magnetic resonance (MR) tagging.
Although the implantation methods have been used for the LV motion analysis, and provide accurate localization, the invasive nature of the procedures does not allow a sufficient number of markers to be implanted for describing the LV geometry. Moreover, it poses the potential problem of local myocardium property alteration due to the introduction of foreign objects. On the other hand, the methods which utilize naturally occurring landmarks, like bifurcations of coronary arteries, do not require a surgery and can provide potentially many markers. However, intra-coronary injection of contrast medium is usually required to highlight the blood vessels in acquired images. When the blood supply is severely obstructed due to arterial occlusion, the tracing of the feature points around the region can be very difficult to achieve.
MR tagging has its advantages over the aforementioned approaches because a large number of material points may be marked and tracked during systole in a non-invasive manner. By locally perturbing the magnetization in tissue, one can create spatially encoded patterns such as starbursts or grids. Those patterns or magnetization tags are seen as dark regions in subsequent images (within a certain relaxation time T.sub.1). As magnetization moves with tissue, the magnetization tags will move in the corresponding images, directly reflecting the motion of the underlying tissue and allowing us to follow the motion patterns within otherwise featureless structures such as the heart wall. One drawback of current MR tagging techniques, is that the tracking is possible only during systole or diastole at one time (i.e., not for a complete heart cycle), due to the decay of the magnetization signal over time.
Recently, curvature-based point correspondence recovery techniques have been proposed by researchers as an alternative to the above methods. One method employs the computation of the Gaussian curvature of a model that deforms based on the assumption of conformal motion. Another method utilizes the potential energy of their bending and stretching model to estimate the curvature. A third method combines curvature extraction with Phase Velocity MRI, in an attempt to assess the transmural myocardial deformation in 2D. A fourth method demonstrates the stability of the Gaussian curvature computation in an experiment where the Gaussian curvature was computed through an iterative relaxation scheme from voxel-based surface rendering of CT left-ventricle volumetric data over a cardiac cycle. The derivation of point-correspondences based on curvature may be applied to data sets from many different medical imaging modalities, and it may provide point-correspondence over an entire heart cycle.
What is needed is an apparatus and method for dynamic modeling, shape estimation, and motion analysis of an object having material points therein. Such an invention can be particularly useful in the dynamic modeling of, for example, the human heart and can provide a visually perceptible representation of the object which can be intuitively understood by an observer of the display, such as a clinician, and can provide quantitative data regarding the object, for example, the nature and extent of a myocardial defect.